Nuprl Lemma : perm_inverse
∀[T:Type]. ∀[p:Perm(T)]. ((p O inv_perm(p) = id_perm() ∈ Perm(T)) ∧ (inv_perm(p) O p = id_perm() ∈ Perm(T)))
Proof
Definitions occuring in Statement :
comp_perm: comp_perm,
inv_perm: inv_perm(p),
id_perm: id_perm(),
perm: Perm(T),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
all: ∀x:A. B[x],
perm: Perm(T),
prop: ℙ,
pi2: snd(t),
perm_b: p.b,
pi1: fst(t),
perm_f: p.f,
mk_perm: mk_perm(f;b),
comp_perm: comp_perm,
inv_perm: inv_perm(p),
id_perm: id_perm(),
inv_funs: InvFuns(A;B;f;g),
true: True,
tidentity: Id{T},
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
perm_wf,
comp_perm_wf,
inv_perm_wf,
perm_properties,
inv_funs_wf,
perm_f_wf,
perm_b_wf,
perm_sig_wf,
mk_perm_wf,
identity_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
universeIsType,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isect_memberEquality,
isectElimination,
because_Cache,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
functionEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[p:Perm(T)]. ((p O inv\_perm(p) = id\_perm()) \mwedge{} (inv\_perm(p) O p = id\_perm()))
Date html generated:
2019_10_16-PM-00_59_03
Last ObjectModification:
2018_09_26-PM-08_09_11
Theory : perms_1
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